Approximaths

Anharmonic oscillator I

Written by

approximaths

in

We would like to illustrate the use of Padé approximants in the context of the anharmonic oscillator. A harmonic oscillator is an oscillating system that experiences a restoring force. Anharmonicity is the deviation of a system from being a harmonic oscillator. The anharmonic oscillator is described by the following differential equation:

\displaystyle \left(-\frac{d^2}{dx^2} + x^2 + x^4\right) \phi(x) = E_n \phi(x)

This differential equation is very hard to solve exactly. In order to obtain an approximate solution, we can use perturbation theory. By inserting a small dimensionless parameter \epsilon, the equation becomes:

\displaystyle \left(-\frac{d^2}{dx^2} + x^2 + \epsilon x^4\right) \phi(x) = E_n(\epsilon) \phi(x)

If we are interested in the energy levels, the basic idea is to write E_n(\epsilon) as a geometric series. For the ground state E_0 we write:

\displaystyle E_0(\epsilon) = \sum_{n=0}^{\infty} E_{0,n} \epsilon^n

The solution is a divergent perturbation series (C.M. Bender and T.T. Wu, Phys. Rev. 184, 1969):

\displaystyle E_0(\epsilon) = \frac{1}{2} + \frac{3}{4}\epsilon - \frac{21}{8}\epsilon^2 + \frac{333}{16}\epsilon^3 + \mathcal{O}(\epsilon^4)

Let’s calculate the P(1,1) approximant of E_0 using the techniques presented in post Computing Padé approximants:

\displaystyle P(1,1) = \frac{A_0 + A_1 \epsilon}{1 + B_1 \epsilon} = C_0 + C_1 \epsilon + C_2 \epsilon^2

Solving the linear systems for computing Padé approximants sequentially leads to:

\displaystyle C_1 B_1 = - C_2 \implies B_1 = -\frac{C_2}{C_1}
\displaystyle \frac{3}{4} B_1 = \frac{21}{8} \implies B_1 = \frac{84}{24} = \frac{7}{2}
\displaystyle \begin{pmatrix} C_0 & 0 \\ C_1 & C_0 \\ \end{pmatrix} \begin{pmatrix} 1 \\ B_1 \end{pmatrix} = \begin{pmatrix} A_0 \\ A_1 \end{pmatrix}
\displaystyle \begin{pmatrix} \frac{1}{2} & 0 \\ \frac{3}{4} & \frac{1}{2} \\ \end{pmatrix} \begin{pmatrix} 1 \\ \frac{7}{2} \end{pmatrix} = \begin{pmatrix} \frac{1}{2} \\ \frac{3}{4} + \frac{7}{4} \end{pmatrix}
\displaystyle P(1,1)_{E_0} = \frac{\frac{1}{2} + \left(\frac{3}{4} + \frac{7}{4}\right)\epsilon}{1 + \frac{7}{2}\epsilon}

Setting \epsilon = 1 (to recover the initial differential equation) we have:

\displaystyle P(1,1)_{E_0} = \frac{\frac{1}{2} + \frac{3}{4} + \frac{7}{4}}{1 + \frac{7}{2}} = 0.6666

The P(4,4)_{E_0} = 1.3838 and the exact solution is 1.3924.

The relative error (definition here) of the approximant P(4,4)_{E_0} is:

\displaystyle \text{Relative error} = \frac{1.3838 - 1.3924}{1.3924} = -0.0062

It’s therefore interesting to note that in order to solve a very difficult differential equation, we can use perturbation theory. In the case of the anharmonic oscillator, the perturbative series is divergent. Using Padé approximants, we can make use of a divergent perturbative series and approximate the exact answer arbitrarily.

Comments

Leave a comment

More posts